Let us start by writing the integral equation for the vector potential
where we presume that we've already transformed the IWE into
the IHE. We will choose to use the outgoing wave Green's function to
make it clear that the field we are looking for is the one that the
source is emitting, not one that it is absorbing.

(11.68)

There is no inhomogeneous term if there are no boundaries with a
priori known boundary conditions.

Note that a more general solution would be one that allowed for
absorption of incoming waves as well as the emission of outgoing waves,
but that this would require knowing something about the sources outside
the domain considered to be infinite. We will talk about this later
(scattering theory and the optical theorem).

From
we can easily find
or
:

(11.69)

(by definition). Outside of the source, though (where the currents are
all zero) Ampere's law tells us that:

(11.70)

or

(11.71)

(11.72)

or

(11.73)

Doing the integral above can be quite difficult in the general case.
However, we'll find that for most reasonable, physical situations
we will be able to employ certain approximations that will enable us to
obtain a systematic hierarchy of descriptions that converge to the
correct answer as accurately as you like, at the same time they increase
our physical insight into the radiative processes.